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Geometriae Dedicata

, Volume 198, Issue 1, pp 149–170 | Cite as

Injective homomorphisms of mapping class groups of non-orientable surfaces

  • Elmas Irmak
  • Luis ParisEmail author
Original Paper
  • 57 Downloads

Abstract

Let N be a compact, connected, non-orientable surface of genus \(\rho \) with n boundary components, with \(\rho \ge 5\) and \(n \ge 0\), and let \(\mathcal M(N)\) be the mapping class group of N. We show that, if \(\mathcal G\) is a finite index subgroup of \(\mathcal M(N)\) and \(\varphi : \mathcal G\rightarrow \mathcal M(N)\) is an injective homomorphism, then there exists \(f_0 \in \mathcal M(N)\) such that \(\varphi (g) = f_0 g f_0^{-1}\) for all \(g \in \mathcal G\). We deduce that the abstract commensurator of \(\mathcal M(N)\) coincides with \(\mathcal M(N)\).

Mathematics Subject Classification

57N05 

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Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsBowling Green State UniversityBowling GreenUSA
  2. 2.IMB, UMR 5584, CNRSUniv. Bourgogne Franche-ComtéDijonFrance

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